Part 1:
The house will likely constitute 'matrimonial property' and, in any divorce settlement, that property will be divided 'fairly' by the court. What is deemed a fair split will depend on all the relevant facts and circumstances.
You may have paid all the bills but why was this? Does your husband look after the children and, following separation, will the children remain with you or your husband? These factors (amongst many others perhaps) will affect what the court considers to be a fair split - which is not neccessarily 50:50 - and in most cases won't be a 50:50 split.
If your husband has possession of the house and you move out and find somewhere else to live, you will have proven his case for him that you don't need that house to live in, because you have demonstrated that you can, and have, found somewhere else to live, and he hasn't. Therefore the most likely outcome is that either he would have to purchase your share of the house from you or, the court makes an order for a division of sale - the house is sold and you get what the court deems to be a fair share. But, and this is very important, there might be a sting in the tail.